Birmingham midfielder Charlee Adams has signed a new one-year contract with the club.
Southampton are confident Jay Rodriguez will commit to a new long-term deal but the player will consider his options.
The 41-year-old moves across Lancashire from Preston, where he was working as head of youth development having called time on a professional career spanning over 1,000 games at the end of last season.
Alexander replaces Micky Mellon in the Highbury Stadium hotseat and is taking on a first senior management role with the npower League Two club.
He guided North End for five games last term alongside David Unsworth in a caretaker capacity before Graham Westley's mid-season arrival.
Despite leading Fleetwood to two promotions in the past three years and currently seeing them occupy the final League Two play-off spot, Mellon was sacked in the wake of last weekend's FA Cup second-round defeat to Aldershot - the Cod Army's third consecutive loss in all competitions.
Alexander's first game at the helm comes on Saturday at home to fellow promotion-chasers Southend.
After agreeing a rolling-contract at Fleetwood, Alexander admitted the prospect of beginning his managerial career with a club boasting evident potential was too good to turn down.
"The chairman approached Preston North End and asked for permission to speak to me, obviously that was relayed to me and I had to weigh it up," he said.
"It didn't take long. I wanted the opportunity to come and speak - I think it would have been very unprofessional if I hadn't have done that because you've got to look at every opportunity along the way.
"We had a great chat for probably two, two-and-a-half hours on Tuesday, went through everything they would expect from a new manager and everything that I would do coming into a club.
"It was a great opportunity, one I had to take and one I'm very pleased to take."
Chairman Andy Pilley revealed glowing references received on Alexander from across the game helped to make up his mind days after unexpectedly relieving Mellon of his duties.
"It's fair to say we had an enormous amount of quality applicants," he said. "We filtered those out down to a few that we were really interested in speaking to.
"We did a lot of research and a lot of diligence on the people that we considered to be serious candidates and there was a consistent theme that everyone we spoke to in the game came back and said that Graham is an outstanding man, a prime example of how to be a professional footballer.
"He squeezed every single bit of success and ability out of his playing career and he's someone that excites us.
"If he can replicate the same with our squad, impose his high standards and his professionalism into this football club then it can only be of great benefit to the football club."